A-level Mathematics/AQA/MFP3

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Series and limits[edit]

Two important limits:

for any real number k

for all k > 0

The basic series expansions[edit]

Improper intergrals[edit]

The integral : is said to be improper if

  1. the interval of integration is infinite, or;
  2. f(x) is not defined at one or both of the end points x=a and x=b, or;
  3. f(x) is not defined at one or more interior points of the interval .

Polar coordinates[edit]

A diagram illustrating the relationship between polar and Cartesian coordinates.

The area bounded by a polar curve[edit]

For the curve

r must be defined and be non-negative throughout the interval

Numerical methods for the solution of first order differential equations[edit]

Euler's formula[edit]

The mid-point formula[edit]

The improved Eular formula[edit]



Second order differential equations[edit]

Euler's identity[edit]

When substituting into the identity gives


Further reading[edit]

The AQA's free textbook [1]